Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Pseudodifferential Models for Ultrasound Waves with Fractional Attenuation

Sebastian Acosta, Jesse Chan, Raven Johnson, Benjamin Palacios

TL;DR

This work develops a pseudodifferential framework for efficient high-frequency ultrasound wave propagation in media with fractional attenuation. By factorizing the wave operator into forward and backward Dirichlet-to-Neumann maps and deriving the leading and higher-order symbols $\lambda^{\pm}$, the authors build wide-angle Padé approximants to construct one-way and two-way sweeping schemes that capture transmission, reflection, and power-law damping. They prove an error bound showing the relative error decays as $\mathcal{O}(\omega^{-1})$ with increasing frequency and Padé order $M$, under mild propagation-angle assumptions, and validate the approach with a proof-of-concept numerical experiment. The method promises a computationally efficient alternative to full-wave methods for biomedical ultrasound, with clear directions for extending to discontinuous media and more advanced discretizations.

Abstract

To strike a balance between modeling accuracy and computational efficiency for simulations of ultrasound waves in soft tissues, we derive a pseudodifferential factorization of the wave operator with fractional attenuation. This factorization allows us to approximately solve the Helmholtz equation via one-way (transmission) or two-way (transmission and reflection) sweeping schemes tailored to high-frequency wave fields. We provide explicitly the three highest order terms of the pseudodifferential expansion to incorporate the well-known square-root first order symbol for wave propagation, the zeroth order symbol for amplitude modulation due to changes in wave speed and damping, and the next symbol to model fractional attenuation. We also propose wide-angle Pade approximations for the pseudodifferential operators corresponding to these three highest order symbols. Our analysis provides insights regarding the role played by the frequency and the Pade approximations in the estimation of error bounds. We also provide a proof-of-concept numerical implementation of the proposed method and test the error estimates numerically.

Pseudodifferential Models for Ultrasound Waves with Fractional Attenuation

TL;DR

This work develops a pseudodifferential framework for efficient high-frequency ultrasound wave propagation in media with fractional attenuation. By factorizing the wave operator into forward and backward Dirichlet-to-Neumann maps and deriving the leading and higher-order symbols , the authors build wide-angle Padé approximants to construct one-way and two-way sweeping schemes that capture transmission, reflection, and power-law damping. They prove an error bound showing the relative error decays as with increasing frequency and Padé order , under mild propagation-angle assumptions, and validate the approach with a proof-of-concept numerical experiment. The method promises a computationally efficient alternative to full-wave methods for biomedical ultrasound, with clear directions for extending to discontinuous media and more advanced discretizations.

Abstract

To strike a balance between modeling accuracy and computational efficiency for simulations of ultrasound waves in soft tissues, we derive a pseudodifferential factorization of the wave operator with fractional attenuation. This factorization allows us to approximately solve the Helmholtz equation via one-way (transmission) or two-way (transmission and reflection) sweeping schemes tailored to high-frequency wave fields. We provide explicitly the three highest order terms of the pseudodifferential expansion to incorporate the well-known square-root first order symbol for wave propagation, the zeroth order symbol for amplitude modulation due to changes in wave speed and damping, and the next symbol to model fractional attenuation. We also propose wide-angle Pade approximations for the pseudodifferential operators corresponding to these three highest order symbols. Our analysis provides insights regarding the role played by the frequency and the Pade approximations in the estimation of error bounds. We also provide a proof-of-concept numerical implementation of the proposed method and test the error estimates numerically.
Paper Structure (8 sections, 3 theorems, 64 equations, 3 figures, 5 tables, 1 algorithm)

This paper contains 8 sections, 3 theorems, 64 equations, 3 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Pseudodifferential factorization of the wave equation
  4. Sweeping method for the Helmholtz equation
  5. Modeling error analysis
  6. Numerical experiments
  7. Conclusion and limitations
  8. Padé approximations

Key Result

Lemma 5.1

\newlabelThm.Lemma10 Under the assumption on the domain $\Omega$, the wave speed $c$, the damping coefficient $a$ and the fractional attenuation coefficient $a_{\alpha}$ stated above, there is a constant $C>0$ independent of $\omega$, such that for all $f\in L^2(\Omega)$, where $v$ is a solution to Helmholtz_eq. The same holds for the (formal) adjoint $\mathcal{H}^*$.

Figures (3)

  • Figure 1: Support in Fourier space (shaded in red) of the solution $u$ in order for the sweeping method \ref{['Eqn.Sweep0']}-\ref{['Eqn.Sweep']} to be accurate. These conditions are stated in Assumption \ref{['supp_fourier']} and employed in Theorem \ref{['Thm.theorem1']}. In physical terms, these conditions ensure that the waves, induced by the source $f$, oscillate in space at least as fast as they oscillate in time, and simultaneously that the $x$-axis is the dominant direction of propagation.
  • Figure 1: (a) Wavespeed profile with an inclusion defined by \ref{['Eqn.WS']} and (b) absorbing sponge defined by \ref{['Eqn.Sponge']} to mitigate effects of the top and bottom boundaries.
  • Figure 2: Comparison between the one-way and two-way numerical solutions. These solutions were computed using Padé approximations of the pseudodifferential symbols with $4$ terms. The frequency is $\omega = 120 \pi$ which fits $60$ wavelengths across the domain. The two-way solution captures the reflections induced by the inclusion. These reflections are visible on plot of the real part and amplitude of the solution, as well as in the Fourier plots.

Theorems & Definitions (5)

  • Lemma 5.1
  • Proof 1
  • Lemma 5.2
  • Proof 2
  • Theorem 5.4